Orthogonal and Unitary Matrices Defined

Hello! Let's dive into our next topic.

In our previous lessons, we've spent a lot of time working with orthogonal vectors and orthonormal bases. We saw how an orthonormal basis simplifies computations like finding a vector's coordinates or projecting it onto a subspace. Today, we're going to explore matrices that are built from these special bases.

This lesson focuses on a class of matrices that represent transformations preserving geometric structure. Our learning outcome is to define and identify orthogonal (real) and unitary (complex) matrices. These matrices are fundamental in both of your areas of interest: they represent rigid transformations like rotations in deep learning applications (e.g., computer graphics, data analysis) and are essential for describing the evolution of quantum states in quantum mechanics.

1. Orthogonal Matrices: Rotations and Reflections in $R^{n}$

We'll start in the familiar territory of real vector spaces. An orthogonal matrix is a special kind of square matrix whose inverse is incredibly easy to compute.

Let's watch a short video that provides the core definition and the main insight into what makes these matrices special.

Orthogonal Matrices - Explained

This video from DataMListic quickly defines an orthogonal matrix and connects the definition to the properties of its column vectors.

Watch from 0:35 to 1:38. Focus on two key points: The algebraic definition: Q^T Q = I. What this definition implies about the column vectors of Q.

As the video explains, a square matrix $Q$ is orthogonal if its transpose is also its inverse:
$Q^{T} Q = I \Leftrightarrow Q^{- 1} = Q^{T}$
This is a remarkable property. Calculating a matrix inverse is computationally expensive, but finding a transpose is trivial.

The Column Vector Perspective

The real magic happens when we consider what the condition $Q^{T} Q = I$ means for the columns of $Q$ . Let the columns of $Q$ be the vectors $q_{1}, q_{2}, \dots, q_{n}$ .

The matrix product $Q^{T} Q$ has entries $(Q^{T} Q)_{ij} = q_{i}^{T} q_{j}$ , which is just the dot product $q_{i} \cdot q_{j}$ . Since $Q^{T} Q$ must be the identity matrix $I$ , we have:
$q_{i} \cdot q_{j} = {10 if i = j if i \neq = j$
This is the definition of an orthonormal set!

So, we have an equivalent and more intuitive definition:

A square matrix $Q$ is orthogonal if and only if its column vectors form an orthonormal basis.

This provides a direct link to our previous work with the Gram-Schmidt process. If you take any basis, apply Gram-Schmidt to get an orthonormal basis, and then form a matrix with these orthonormal vectors as its columns, you have just created an orthogonal matrix.

A Note on Terminology

You might be wondering why we call them "orthogonal matrices" when their columns are actually "orthonormal." This is a common point of confusion. For a deeper explanation and to see some classic examples, let's turn to a reading.

Orthogonal Matrix: An Explanation with Examples and Code

This article from DataCamp provides a clear definition, concrete examples, and an excellent clarification on the 'orthogonal' vs. 'orthonormal' naming convention.

Please read the following three sections: 'What Is an Orthogonal Matrix?': This reinforces the definition. 'Examples of Orthogonal Matrices': Pay attention to the rotation and reflection matrix examples. These are the archetypes of orthogonal transformations. 'Orthogonal vs. Orthonormal Matrices': This section directly addresses the naming convention.

The key takeaway is that "orthogonal matrix" is the standard, historical term, and it always implies that the columns (and rows) are orthonormal.

Definitions and Examples of Orthogonal and Unitary Matrices — This image provides a concise summary of the definitions for both orthogonal and unitary matrices, along with a simple 2x2 example for each. We will cover the unitary matrix definition next.

How to Identify an Orthogonal Matrix

Given a square matrix $Q$ , to check if it's orthogonal, you simply compute the product $Q^{T} Q$ and see if the result is the identity matrix $I$ .

Test your understanding!

Is the following matrix orthogonal?
$A = \frac{1}{2} (11 1 - 1)$

Show answer

Let's compute $A^{T} A$ .
First, find the transpose $A^{T}$ :
$A^{T} = \frac{1}{2} (11 1 - 1)^{T} = \frac{1}{2} (11 1 - 1)$
In this case, the matrix is symmetric, so $A^{T} = A$ .

Now, multiply $A^{T} A$ :
$A^{T} A = (\frac{1}{2})^{2} (11 1 - 1) (11 1 - 1)$
$A^{T} A = \frac{1}{2} ((1) (1) + (1) (1) (1) (1) + (- 1) (1) (1) (1) + (1) (- 1) (1) (1) + (- 1) (- 1))$
$A^{T} A = \frac{1}{2} (2002) = (1001) = I$
Since $A^{T} A = I$ , the matrix $A$ is indeed orthogonal.

2. Unitary Matrices: The Complex Analogue

For quantum mechanics, we must work with complex vector spaces. The concept of an orthogonal matrix generalizes beautifully to the complex domain, but we need to adjust our definition of the "transpose" and the inner product.

The Conjugate Transpose

As you know from your background, a complex number $z = a + bi$ has a conjugate $z ˉ = a - bi$ . When we work with complex matrices, the direct analogue of the transpose is the conjugate transpose, also called the Hermitian transpose or adjoint.

To find the conjugate transpose of a matrix $A$ , denoted $A^{†}$ (read "A dagger") or $A^{H}$ , you first take the transpose of $A$ and then take the complex conjugate of every entry.
$(A^{†})_{ij} = A_{j i}$
For example, if $A = (1 + i 3 2 4 - 5 i)$ , then $A^{†} = (1 - i 2 3 4 + 5 i)$ .

Let's watch a video that introduces this operation and sets the stage for defining unitary matrices.

Complex, Hermitian, and Unitary Matrices

Professor Dave Explains provides a clear introduction to complex matrices and the crucial operation of the conjugate transpose.

Watch the first five minutes of the video (0:00 - 5:01). It covers complex matrices, the conjugate of a matrix, and finally the definition and motivation for the conjugate transpose (M^H or M^\dagger).

Definition of a Unitary Matrix

With the conjugate transpose in hand, the definition of a unitary matrix is a perfect parallel to that of an orthogonal matrix.

A square complex matrix $U$ is unitary if its conjugate transpose is also its inverse:
$U^{†} U = I \Leftrightarrow U^{- 1} = U^{†}$

Just as before, this definition implies that the columns of a unitary matrix form an orthonormal basis, but this time with respect to the complex inner product $⟨ v, w ⟩ = v^{†} w$ .

A crucial point:

If a unitary matrix happens to contain only real numbers, its conjugate transpose is the same as its regular transpose ( $U^{†} = U^{T}$ ). Therefore, a real unitary matrix is simply an orthogonal matrix.

The following resource provides a formal definition and elaborates on this relationship.

Unitary matrix

This page from Grokipedia formally defines unitary matrices and directly compares them to their real counterparts, the orthogonal matrices.

Read the sections titled 'Definition', 'Relation to Orthogonal Matrices', and 'Column and Row Orthogonality'. This will solidify the formal definition and its connection to orthonormality.

How to Identify a Unitary Matrix

The process is identical to the orthogonal case, but using the conjugate transpose. Given a square complex matrix $U$ , compute $U^{†} U$ and check if it equals the identity matrix.

Let's see this in action.

Complex, Hermitian, and Unitary Matrices

Let's return to the video from Professor Dave Explains, where he now defines unitary matrices and works through a concrete example.

Watch from 5:48 to 7:50. He defines a unitary matrix and verifies a 2x2 complex matrix is unitary by calculating U^H U.

Test your understanding!

Is the following matrix unitary?
$U = \frac{1}{2} (1 i i 1)$

Show answer

Let's compute $U^{†} U$ .
First, find the conjugate transpose $U^{†}$ . We transpose the matrix and conjugate its elements.
$U^{T} = \frac{1}{2} (1 i i 1) ⟹ U^{†} = \frac{1}{2} (1 ˉ i ˉ i ˉ 1 ˉ) = \frac{1}{2} (1 - i - i 1)$
Now, multiply $U^{†} U$ :
$U^{†} U = (\frac{1}{2})^{2} (1 - i - i 1) (1 i i 1)$
$U^{†} U = \frac{1}{2} ((1) (1) + (- i) (i) (- i) (1) + (1) (i) (1) (i) + (- i) (1) (- i) (i) + (1) (1))$
Remember that $i^{2} = - 1$ .
$U^{†} U = \frac{1}{2} (1 - (- 1) - i + i i - i - (- 1) + 1) = \frac{1}{2} (2002) = (1001) = I$
Since $U^{†} U = I$ , the matrix $U$ is unitary.

Conclusion

Today we've introduced two extremely important classes of matrices that serve as the foundation for describing rigid motions and transformations in real and complex spaces.

Key Takeaways:

An orthogonal matrix $Q$ is a real square matrix whose columns form an orthonormal basis. This is equivalent to the algebraic condition $Q^{T} Q = I$ , meaning $Q^{- 1} = Q^{T}$ .
A unitary matrix $U$ is a complex square matrix whose columns form an orthonormal basis with respect to the complex inner product. This is equivalent to $U^{†} U = I$ , meaning $U^{- 1} = U^{†}$ .
The procedure for identifying these matrices is to multiply the matrix by its (conjugate) transpose and check if the result is the identity matrix.

Preview of the Next Lesson:

We've defined these matrices algebraically and connected them to orthonormal bases. But what do they do geometrically? In our next lesson, we will prove the most important property of these matrices: they preserve norms and inner products. This means that when you apply an orthogonal or unitary matrix to a set of vectors, their lengths and the angles between them do not change. This is why they perfectly represent rotations and reflections, and why they are essential for describing the evolution of physical systems in quantum mechanics where total probability must be conserved.

Recommended resources

1. Orthogonal Matrices: Rotations and Reflections in Rn